It's taking forever. I played Pokemon Amie tons of times just to evolve it and cause its fun. Its affection level is at its highest and its level 66, my highest level Pokemon. (Which I find pretty shocking but only a little bit.)

First, you'll need to teach your Eevee a fairy-type move, but that's not enough! You also need to make it very, very happy. The solution lies in the new Pokemon-Amie system, which allows you to play and pet your Pokemon. In order to evolve to Sylveon, you're going to need to have all three sections in Amie--Affection, Fullness, and Enjoyment--up to max. Filling up Enjoyment is as easy as playing a bunch of minigames, and that will give you enough snacks to make sure your Pokemon's Fullness is max.

I've actually done it with max Affection and Fullness only, with the Enjoyment being at around 1-2, though the Fairy-move is definitely required. Earliest level-up (after catching one on Route 10) is with Charm, at Lv 29.

It's taking forever. I played Pokemon Amie tons of times just to evolve it and cause its fun. Its affection level is at its highest and its level 66, my highest level Pokemon. (Which I find pretty shocking but only a little bit.)

So I bred my eevees. Equipping one of my resulting younglings with a sooth bell, I promptly started maxing out her poke amie stats. Once done (eevee still at level 1), training her on route 19 with the use of exp share, she evolved promptly at lv 13 after learning baby doll eyes. I'm not certain if the sooth bell is a necessity, but the baby doll eyes skill certainly is.

Yeah, you only need to max out Affection in Pokemon Amie (although Fullness pretty much goes along with that), but it's a pretty easy cycle. Pet Eevee until she stops giving out hearts and starts giving out musical notes, then feed Eevee until she can't eat any more, then play the Head Bounce game on Unlimited mode (get at least three stars on Hard to unlock). As long as you score 200+ in it you should be able to repeat the cycle of petting and feeding Eevee, and it will take about 20 minutes or so to max out Affection.

Maxing out Affection is an awesome move for any Pokemon tbh. They get boosted Exp, can often avoid enemy attacks or status ailments, sometimes pull out surprise survivals at 1HP, and get critical hits way more often. You can even do more than one Pokemon at a time in Amie by petting and feeding them all, playing a game of Head Bounce and then repeating.

The jury's still out on the exact ratio of Fullness, Affection, and Enjoyment you need, but one thing that's definitely key to getting Sylveon is Pokémon Amie; the mascot Eevee-lution of the Gen needs it and a Fairy Type move to evolve. I bred an Eevee with Baby Doll Eyes as an Egg Move and managed to get my Sylveon when it first leveled (hit level three).

I personally max Affection to be safe, but I have seen other people's Sylveons with two or three hearts. I can't say that you need max Fullness and Enjoyment though, because my Sylveon had neither when she evolved.

Fun fact: Sylveon only gets STAB attack moves at level 9, level 20, and from what I hear level 35-36. They are Fairy Wind (9), Draining Kiss (20), and Moonblast (mid/late 30's). If you wait to get Charm or catch and level your Eevee too much, you're missing out on opportunities for it to dish out STAB damage for a while. Jus' saying.